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Abstract
Carathéodory’s axiomatic development of thermodynamics is applied here to the thermohaline circulation of the ocean. The helicity of the differential form
Abstract
Carathéodory’s axiomatic development of thermodynamics is applied here to the thermohaline circulation of the ocean. The helicity of the differential form
Abstract
A parallel algorithm is described for variational assimilation of observations into oceanic and atmospheric models. The algorithm may be coded first for execution on a serial computer and then trivially modified for execution on a parallel computer such as the Intel iPSC/860. The speedup factor for parallel execution is roughly P(2M+3) (2M+3P)−1, where P is the number of processors and M is the number of observations (M≥P). The speedup factor approaches P from below as M→∞.
The algorithm has been applied in serial form to ocean tides (Bennett and McIntosh 1982; McIntosh and Bennett 1984; Bennett 1985) and oceanic equatorial interannual variability (Bennett 1990). It has been applied in parallel form to oceanic synoptic-scale circulation (Bennett and Thorburn 1992); a parallel application to operational forecasting of tropical cyclones is in progress (Bennett et al. 1992).
For the sake of simplicity, the parallel algorithm is described here for a model consisting of a linear, first-order wave equation with initial and boundary conditions plus a dataset consisting of observations at isolated points in space and time. However, measurements of parallel performance are given for a nonlinear quasigeostrophic model.
Abstract
A parallel algorithm is described for variational assimilation of observations into oceanic and atmospheric models. The algorithm may be coded first for execution on a serial computer and then trivially modified for execution on a parallel computer such as the Intel iPSC/860. The speedup factor for parallel execution is roughly P(2M+3) (2M+3P)−1, where P is the number of processors and M is the number of observations (M≥P). The speedup factor approaches P from below as M→∞.
The algorithm has been applied in serial form to ocean tides (Bennett and McIntosh 1982; McIntosh and Bennett 1984; Bennett 1985) and oceanic equatorial interannual variability (Bennett 1990). It has been applied in parallel form to oceanic synoptic-scale circulation (Bennett and Thorburn 1992); a parallel application to operational forecasting of tropical cyclones is in progress (Bennett et al. 1992).
For the sake of simplicity, the parallel algorithm is described here for a model consisting of a linear, first-order wave equation with initial and boundary conditions plus a dataset consisting of observations at isolated points in space and time. However, measurements of parallel performance are given for a nonlinear quasigeostrophic model.
Abstract
Numerical simulations of two-dimensional turbulence show that O(κ−1) and O(κ−4) energy spectra—described by Fox and Orszag (1973a) as enstrophy-equipartitioning and strongly dissipating turbulence, respectively—occur independently of the type of dissipation mechanism, and the inclusion of forcing and the β-effect. In both states the modal decorrelation rate η depends strongly upon wavenumber in accordance with the equations of a direct interaction approximation (Kraichnan, 1958), but in conflict with the hypothetical wavenumber independence of η in an enstrophy-caseading inertial similarity range. Implications for geophysical fluid dynamical modeling are discussed.
Abstract
Numerical simulations of two-dimensional turbulence show that O(κ−1) and O(κ−4) energy spectra—described by Fox and Orszag (1973a) as enstrophy-equipartitioning and strongly dissipating turbulence, respectively—occur independently of the type of dissipation mechanism, and the inclusion of forcing and the β-effect. In both states the modal decorrelation rate η depends strongly upon wavenumber in accordance with the equations of a direct interaction approximation (Kraichnan, 1958), but in conflict with the hypothetical wavenumber independence of η in an enstrophy-caseading inertial similarity range. Implications for geophysical fluid dynamical modeling are discussed.
Abstract
The ill-posedness of regional primitive equation models is examined, using a regional shallow-water model. The ill-posedness is resolved by reformulation as a least-squares inverse problem, in the sense that the Euler-Lagrange or variational boundary conditions ensure unique solutions for the linearized problem. The inverses for nonlinear problems are calculated using variants of simulated annealing and massively parallel computing. Simple experiments compare the relative merits of pointwise measurements and path-integrated measurements in compensating for bad boundary data. Error statistics are calculated, despite the large dimension of the state space.
Abstract
The ill-posedness of regional primitive equation models is examined, using a regional shallow-water model. The ill-posedness is resolved by reformulation as a least-squares inverse problem, in the sense that the Euler-Lagrange or variational boundary conditions ensure unique solutions for the linearized problem. The inverses for nonlinear problems are calculated using variants of simulated annealing and massively parallel computing. Simple experiments compare the relative merits of pointwise measurements and path-integrated measurements in compensating for bad boundary data. Error statistics are calculated, despite the large dimension of the state space.
Abstract
An analysis of variational data assimilation schemes for linear dynamical forecast models shows that the penalty functional must include an explicit contribution from the initial conditions in order to ensure a unique, low-noise forecast. The noise level is related to the effective number of data being assimilated.
Abstract
An analysis of variational data assimilation schemes for linear dynamical forecast models shows that the penalty functional must include an explicit contribution from the initial conditions in order to ensure a unique, low-noise forecast. The noise level is related to the effective number of data being assimilated.
Abstract
The generalized inverse is constructed for a nonlinear, single-layer quasigeostrophic model, together with initial conditions and a finite number of interior data. With the exception of doubly Periodic boundary condition all constraints are weak. The inverse minimizes a penalty functional that is quadratic in the errors in prior estimates of the model forcing, the initial conditions, and the measurements. The quadratic form consists of the inverses of the prior error covariances. The nonlinear Euler-Lagrange equations are solved iteratively. Each iterate is a linear Euler-Lagrange problem, which in turn is solved in terms of a prior streamfunction estimate, plus a finite-linear combination of representers (one for each linear measurement functional). The sequence of inverse streamfunction estimates is bounded and so has at least one limit Point. The sequence of representer matrices is uniformly positive definite and so the limiting inverse is a minimum, rather than simply an extremum of the penalty functional. The streamfunction estimated with the weak constraint inverse is compared. with the result of a strong constraint inverse, which is identical in every respect, except that the prior estimate of the model forcing is assumed to be exact.
A statistical linearization about the limiting inverse yields the inverse or posterior error covariance. Similarly, the reduced penalty functional is a χ2 variable with as many effective degrees of freedom as the observing system.
The algorithm is easily implemented on multiple-processor computers. The parameter experiments reported here were Performed on a 128-processor Intel iPSC/860 and also on a single-processor workstation. The formal number of degrees of freedom exceeded a quarter of a million. The representer algorithm is tractable and stable since it identifies the five number of degrees of freedom, namely, the number M of measurements. Here M = 128.
Abstract
The generalized inverse is constructed for a nonlinear, single-layer quasigeostrophic model, together with initial conditions and a finite number of interior data. With the exception of doubly Periodic boundary condition all constraints are weak. The inverse minimizes a penalty functional that is quadratic in the errors in prior estimates of the model forcing, the initial conditions, and the measurements. The quadratic form consists of the inverses of the prior error covariances. The nonlinear Euler-Lagrange equations are solved iteratively. Each iterate is a linear Euler-Lagrange problem, which in turn is solved in terms of a prior streamfunction estimate, plus a finite-linear combination of representers (one for each linear measurement functional). The sequence of inverse streamfunction estimates is bounded and so has at least one limit Point. The sequence of representer matrices is uniformly positive definite and so the limiting inverse is a minimum, rather than simply an extremum of the penalty functional. The streamfunction estimated with the weak constraint inverse is compared. with the result of a strong constraint inverse, which is identical in every respect, except that the prior estimate of the model forcing is assumed to be exact.
A statistical linearization about the limiting inverse yields the inverse or posterior error covariance. Similarly, the reduced penalty functional is a χ2 variable with as many effective degrees of freedom as the observing system.
The algorithm is easily implemented on multiple-processor computers. The parameter experiments reported here were Performed on a 128-processor Intel iPSC/860 and also on a single-processor workstation. The formal number of degrees of freedom exceeded a quarter of a million. The representer algorithm is tractable and stable since it identifies the five number of degrees of freedom, namely, the number M of measurements. Here M = 128.
Abstract
Meridional eddy heat flux in the subtropical North Pacific is estimated from TRANSPAC ship-of-opportunity data collected during 1976–80. Two methods are used. The first fits simple functional forms to be temporal anomalies of the temperature field; thermal wind velocities are derived analytically and the presence of vertical phase tilts lead to transient eddy heat fluxes between 140° and 170°E of +0.01 PW at 32°N in particular. This estimate includes contributions from a range of zonal wavenumber.
The second method calculates thermal winds relative to 400 m bv simple differences on the data grid. Transient and stationary eddy heat fluxes between Japan and 120°W are derived: the former are typically +.02 PW while the latter has a maximum of +0.1 PW at 32°N associated with the meander in the Kuroshio Current over the Sititõ Ridge. relative to 1000 m, the stationary eddy heat flux may be as high as 0.4 PW.
Our total eddy heat flux of +0.4 PW, combined with classical hydrographic estimates, is consistent with recent surface budget studies but not with atmospheric residual fluxes.
Abstract
Meridional eddy heat flux in the subtropical North Pacific is estimated from TRANSPAC ship-of-opportunity data collected during 1976–80. Two methods are used. The first fits simple functional forms to be temporal anomalies of the temperature field; thermal wind velocities are derived analytically and the presence of vertical phase tilts lead to transient eddy heat fluxes between 140° and 170°E of +0.01 PW at 32°N in particular. This estimate includes contributions from a range of zonal wavenumber.
The second method calculates thermal winds relative to 400 m bv simple differences on the data grid. Transient and stationary eddy heat fluxes between Japan and 120°W are derived: the former are typically +.02 PW while the latter has a maximum of +0.1 PW at 32°N associated with the meander in the Kuroshio Current over the Sititõ Ridge. relative to 1000 m, the stationary eddy heat flux may be as high as 0.4 PW.
Our total eddy heat flux of +0.4 PW, combined with classical hydrographic estimates, is consistent with recent surface budget studies but not with atmospheric residual fluxes.
Abstract
When averaging the equations of motion, thermodynamics, and scalar conservation over turbulent fluctuations, we perform the process in several stages. First, an average is taken over the microscopic scales of turbulence, including the centimeter-scale band in which the dissipation of kinetic energy and temperature or density variance occurs. The eddy-correlation fluxes that arise in this stage are called microstructure fluxes. Next, the equations are transformed into coordinates relative to the microscopically averaged isopycnals. Finally, an average is taken, relative to these isopycnals, over macroscopic scales of eddy variability, which may include the mesoscale band of planetary motions. Average transport terms, analogous to conventional Reynolds transports in fixed-depth averages, arise also from the macroscopic eddies. This is not so for density, for which no counterparts of macroscopic Reynolds transports exist on constant density surfaces. Only microstructure flux divergence, which is synonymous with diapycnal velocity, contributes to the density balance. Under the assumption that microstructure density variance production is in equilibrium with its molecular dissipation, the microstructure density flux has the form of the molecular flux of heat down the vertical mean gradient, amplified by the Cox number. Munk's abyssal recipe for the vertical velocity/diffusivity ratio can now be reinterpreted as the diapycnal velocity/diffusivity ratio.
Abstract
When averaging the equations of motion, thermodynamics, and scalar conservation over turbulent fluctuations, we perform the process in several stages. First, an average is taken over the microscopic scales of turbulence, including the centimeter-scale band in which the dissipation of kinetic energy and temperature or density variance occurs. The eddy-correlation fluxes that arise in this stage are called microstructure fluxes. Next, the equations are transformed into coordinates relative to the microscopically averaged isopycnals. Finally, an average is taken, relative to these isopycnals, over macroscopic scales of eddy variability, which may include the mesoscale band of planetary motions. Average transport terms, analogous to conventional Reynolds transports in fixed-depth averages, arise also from the macroscopic eddies. This is not so for density, for which no counterparts of macroscopic Reynolds transports exist on constant density surfaces. Only microstructure flux divergence, which is synonymous with diapycnal velocity, contributes to the density balance. Under the assumption that microstructure density variance production is in equilibrium with its molecular dissipation, the microstructure density flux has the form of the molecular flux of heat down the vertical mean gradient, amplified by the Cox number. Munk's abyssal recipe for the vertical velocity/diffusivity ratio can now be reinterpreted as the diapycnal velocity/diffusivity ratio.
Abstract
A nonlinear 2½-layer reduced gravity primitive equations (PE) ocean model is used to assimilate sea surface temperature (SST) data from the Tropical Atmosphere–Ocean (TAO) moored buoys in the tropical Pacific. The aim of this project is to hindcast cool and warm events of this part of the ocean, on seasonal to interannual timescales.
The work extends that of Bennett et al., who used a modified Zebiak–Cane coupled model. They were able to fit a year of 30-day averaged TAO data to within measurement errors, albeit with significant initial and dynamical residuals. They assumed a 100-day decorrelation timescale for the dynamical residuals. This long timescale for the residuals reflects the neglect of resolvable processes in the intermediate coupled model, such as horizontal advection of momentum. However, the residuals in the nonlinear PE model should be relatively short timescale errors in parameterizations. The scales for these residuals are crudely estimated from the upper ocean turbulence studies of Peters et al. and Moum.
The assimilation is performed by minimizing a weighted least squares functional expressing the misfits to the data and to the model throughout the tropical Pacific and for 18 months. It is known that the minimum lies in the “data subspace” of the state or solution space. The minimum is therefore sought in the data subspace, by using the representer method to solve the Euler–Lagrange (EL) system. Although the vector space decomposition and solution method assume a linear EL system, the concept and technique are applied to the nonlinear EL system (resulting from the nonlinear PE model), by iterating with linear approximations to the nonlinear EL system. As a first step, the authors verify that sequences of solutions of linear iterates of the forward PE model do converge. The assimilation is also used as a significance test of the hypothesized means and covariances of the errors in the initial conditions, dynamics, and data. A “strong constraint” inverse solution is computed. However, it is outperformed by the “weak constraint” inverse.
A cross validation by withheld data is presented, as well as an inversion with the model forced by the Florida State University winds, in place of a climatological wind forcing used in the former inversions.
Abstract
A nonlinear 2½-layer reduced gravity primitive equations (PE) ocean model is used to assimilate sea surface temperature (SST) data from the Tropical Atmosphere–Ocean (TAO) moored buoys in the tropical Pacific. The aim of this project is to hindcast cool and warm events of this part of the ocean, on seasonal to interannual timescales.
The work extends that of Bennett et al., who used a modified Zebiak–Cane coupled model. They were able to fit a year of 30-day averaged TAO data to within measurement errors, albeit with significant initial and dynamical residuals. They assumed a 100-day decorrelation timescale for the dynamical residuals. This long timescale for the residuals reflects the neglect of resolvable processes in the intermediate coupled model, such as horizontal advection of momentum. However, the residuals in the nonlinear PE model should be relatively short timescale errors in parameterizations. The scales for these residuals are crudely estimated from the upper ocean turbulence studies of Peters et al. and Moum.
The assimilation is performed by minimizing a weighted least squares functional expressing the misfits to the data and to the model throughout the tropical Pacific and for 18 months. It is known that the minimum lies in the “data subspace” of the state or solution space. The minimum is therefore sought in the data subspace, by using the representer method to solve the Euler–Lagrange (EL) system. Although the vector space decomposition and solution method assume a linear EL system, the concept and technique are applied to the nonlinear EL system (resulting from the nonlinear PE model), by iterating with linear approximations to the nonlinear EL system. As a first step, the authors verify that sequences of solutions of linear iterates of the forward PE model do converge. The assimilation is also used as a significance test of the hypothesized means and covariances of the errors in the initial conditions, dynamics, and data. A “strong constraint” inverse solution is computed. However, it is outperformed by the “weak constraint” inverse.
A cross validation by withheld data is presented, as well as an inversion with the model forced by the Florida State University winds, in place of a climatological wind forcing used in the former inversions.
Abstract
Practical hydrostatic ocean models are often restricted to statically stable configurations by the use of a convective adjustment. A common way to do this is to assign an infinite boat conductivity to the water at a given level if the water column should become statically unstable. This is implemented in the form of a switch. When a statically unstable configuration is detected, it is immediately replaced with a statically stable one in which heat is conserved. In this approach, the model is no longer governed by a smooth set of equations, and usual techniques of variational data assimilation must be modified.
In this note, a simple one-dimensional diffusive model is presented. Despite its simplicity, this model captures the essential behavior of the convective adjustment scheme in a widely used ocean general circulation model. Since this simple model can be derived from the more complex general circulation model, it then follows that many of the properties of the constrained system can be observed in this very simple scalar ordinary differential equation with a constraint on the solution.
Techniques from the theory of optimal control are used to find solutions of a simple formulation of the variational data assimilation problem in this simple case. The optimal solution involves the solution of a nonlinear problem, even when the unconstrained dynamics are linear. In cases with discontinuous dynamics, one cannot define the adjoint of the linearized system in a straightforward manner. The very simplest variational formulation is shown to have nonunique stationary points and undesirable physical consequences. Modifications that lead to better behaved calculations and more meaningful solutions are presented.
Whereas it is likely that the underlying principles from control theory are applicable to practical ocean models, the technique used to solve the simple problem may be applicable only to steady problems. Derivation of suitable techniques for initial value problems will involve a major research effort.
Abstract
Practical hydrostatic ocean models are often restricted to statically stable configurations by the use of a convective adjustment. A common way to do this is to assign an infinite boat conductivity to the water at a given level if the water column should become statically unstable. This is implemented in the form of a switch. When a statically unstable configuration is detected, it is immediately replaced with a statically stable one in which heat is conserved. In this approach, the model is no longer governed by a smooth set of equations, and usual techniques of variational data assimilation must be modified.
In this note, a simple one-dimensional diffusive model is presented. Despite its simplicity, this model captures the essential behavior of the convective adjustment scheme in a widely used ocean general circulation model. Since this simple model can be derived from the more complex general circulation model, it then follows that many of the properties of the constrained system can be observed in this very simple scalar ordinary differential equation with a constraint on the solution.
Techniques from the theory of optimal control are used to find solutions of a simple formulation of the variational data assimilation problem in this simple case. The optimal solution involves the solution of a nonlinear problem, even when the unconstrained dynamics are linear. In cases with discontinuous dynamics, one cannot define the adjoint of the linearized system in a straightforward manner. The very simplest variational formulation is shown to have nonunique stationary points and undesirable physical consequences. Modifications that lead to better behaved calculations and more meaningful solutions are presented.
Whereas it is likely that the underlying principles from control theory are applicable to practical ocean models, the technique used to solve the simple problem may be applicable only to steady problems. Derivation of suitable techniques for initial value problems will involve a major research effort.
